This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers. This talk concerns the question: Which multiplicities are nonzero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. Both appear in representation theory as tensor product multiplicities for a classical Lie group. The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Our methods are bijective, utilizing diagrams called bumpless pipe dreams to obtain combinatorial interpretations of the coefficients c w and c (β) w appearing in these bounds. We then see how to extend this result to principal specializations of β-Grothendieck polynomials ν (β) w := G (β) w(1.,1) in the setting where w is additionally vexillary. In this talk, we discuss a pattern containment formula (originally conjectured by Yibo Gao) which gives a lower bound for ν w whenever w is 1243-avoiding. (Dennin): The principal specialization of a Schubert polynomial ν w := S w(1.,1) is known to give the degree of the matrix Schubert variety corresponding to the permutation w. For Fano problems of moderate size with as yet undetermined Galois group, computational methods are used to prove the Galois group is the full symmetric group. Those Fano problems with finitely many solutions have an associated Galois group that acts on the set of solutions. (Yahl): The problem of enumerating linear spaces of a fixed dimension on a variety is known as a Fano problem. Throughout the talk I will give examples and specific computations. In this expository talk, I will introduce tropical cohomology, the various ways of computing cohomology, and some interesting properties. As a cohomology theory germane to tropical geometry, tropical cohomology is interesting to study in its own right, with analogues of Poincar´ duality and the Lefschetz (1,1)-Theorem. For tropical varieties coming from complex hyperplane arrangement complements, tropical cohomology allows one to compute the de Rham cohomology of the hyperplane arrangement complement in a combinatorial way. (Binder): Tropical homology and cohomology arose as a way to study the Hodge structure of a family of complex projective varieties in passing to a tropical limit. In this talk, we explain how to prove the support conjecture for skew-symmetric rank-2 cluster algebras by applying the BBD decomposition theorem to various morphisms related to Nakajima's graded quiver varieties. The support conjecture in gives a conjectural description of the support of a triangular basis element for any rank-2 cluster algebra. (Li): Berenstein and Zelevinsky introduced quantum cluster algebras and their triangular bases. Torsion points of abelian varieties over torsion fields Splice type surface singularities and their local tropicalizationsĬayley-Bacharach sets and Discriminant complementsĭerived equivalences of gerbes and the arithmetic of genus 1 curvesĬounting parabolic principal G-bundles with nilpotent sections over P 1Ī non-Archimedean characterization of local K-stabilityĪ Gröbner basis for Kazhdan-Lusztig ideals in the flag variety of affine type A The period-index problem over the complex number Quadratic forms, local-global principles, and field invariants Holomorphic anomaly equations for C n/ Z n Gromov-Witten Invariants and Cohomological Field Theories Pattern bounds for principal specializations of β-Grothendieck polynomialsĬolloquium talk: Newell-Littlewood numbersįiltrations on combinatorial intersection cohomology and invariants of subdivisionsĬolloquium talk: Sliding Block Puzzles With a TwistĪ Combinatorial Search for Mori Dream SpacesĬurve classes on conic bundle threefolds and applications to rationality.īrauer group and 0-cycles on smooth varieties Nakajima's graded quiver varieties and the triangular bases of cluster algebras Location: MW 154 (in person) or Zoom (virtual, email the organizers for the Zoom coordinates) Ohio State University Algebraic Geometry Seminar
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